"Stochastic volatility models can be thought of as sticky delta model. And Local volatility model as sticky Strike." Please help me understand how the author has reached this conclusion.

  • $\begingroup$ Where is that from ? LV most certainly is not sticky strike by the commonly accepted definition of sticky strike i.e. IV(k) remains constant whatever the spot does. $\endgroup$ – Ivan Oct 13 '19 at 18:40
  • $\begingroup$ yes you are right. LV is seen close to Sticky strike. I am not able to understand this. $\endgroup$ – Ussu Oct 13 '19 at 18:47
  • $\begingroup$ gaussiandotblog.files.wordpress.com/2018/02/… $\endgroup$ – Ussu Oct 13 '19 at 18:49
  • $\begingroup$ page 395 or search "Dupire". You will be directed to stochastic and local volatility section. $\endgroup$ – Ussu Oct 13 '19 at 18:53
  • $\begingroup$ This is only true for (log-)space homogeneous SV models (e.g. Heston) and concerns the partial derivative Delta $\endgroup$ – Quantuple Oct 14 '19 at 9:59

Intuitively, in a (log)-space homogenous diffusion model $$ S_t \propto S_0, \forall t \geq 0 $$ such that implied volatilities will only depend on the moneyness level and not on the absolute spot level, which is precisely the definition of sticky delta.

Mathematically, consider a (log)-space homogeneous diffusion model (be it stochastic or not) $$ \frac{dS_t}{S_t} = \mu(\cdot) dt + \sigma(\cdot) dW_t,\,\,\,S(0) = S_0 $$ where by (log)-space homogeneous we mean that the drift and diffusion coefficients on the RHS do not involve $S_t$. As such:

  • a LV model is not space homogeneous since $\sigma(\cdot) = \sigma(t,S_t)$
  • a SV model à la Heston is space homogeneous since $\sigma(\cdot) = \sqrt{v_t}$ with $v_t$ given by a separate SDE.

[Homogeneity relationship] Because of (log-)space homogeneity we have that the price of a European vanilla option is a homogeneous function of degree 1 i.e. $$ C(\xi S_0, \xi K, T) = \xi C(S_0, K, T), \forall \xi > 0 $$ such that by Euler's theorem (i.e. taking the derivative of the above wrt to $\xi$ and evaluating it at $\xi = 1$) we get $$ C = \frac{\partial C}{S_0} S_0 + \frac{\partial C}{\partial K} K $$

[IV stickiness (1/2)] Consider a space homogeneous diffusion model with parameters $\Theta$. The corresponding implied volatility surface is the mapping \begin{align} \Sigma &: (S_0, K, T) \to \Sigma(S_0,K,T) \\ \text{such that } & C(S_0,K,T;\Theta) = C_{BS}(S_0, K, T; \Sigma(S_0,K,T)) \end{align} where $C_{BS}(.)$ denotes the Black-Scholes pricing formula for a European call option.

Given the model's space homogeneity, we just showed that: $$ S_0 \frac{\partial C}{\partial S_0}(S_0,K,T;\Theta) + K \frac{\partial C}{\partial K}(S_0,K,T;\Theta) = C(S_0,K,T;\Theta) $$ Plugging in the above implied volatility definition then allows one to write (chain rule) $$ S_0 \left[ \frac{\partial C_{BS}}{\partial S_0}(S_0, K, T; \Sigma) + \frac{\partial C_{BS}}{\partial \Sigma} \frac{\partial \Sigma}{\partial S_0}(S_0,K,T) \right] + K \left[ \frac{\partial C_{BS}}{\partial K}(S_0, K, T; \Sigma) + \frac{\partial C_{BS}}{\partial\Sigma} \frac{\partial \Sigma}{\partial K}(S_0,K,T) \right] = C_{BS}(S_0, K, T; \Sigma) $$

Denoting the Black-Scholes Vega by $\nu$ and noting that the Black-Scholes model is itself space homogeneous, one gets \begin{gather*} S_0 \left[ \frac{\partial C_{BS}}{\partial S_0}(S_0, K, T; \Sigma) + \nu \frac{\partial \Sigma}{\partial S_0}(S_0,K,T) \right] + K \left[ \frac{\partial C_{BS}}{\partial K}(S_0, K, T; \Sigma) + \nu \frac{\partial \Sigma}{\partial K}(S_0,K,T) \right] = C_{BS}(S_0, K, T; \Sigma) \end{gather*} Or equivalently rearranging terms: \begin{gather*} \nu \left[ S_0 \frac{\partial \Sigma}{\partial S_0}(S_0,K,T) + K \frac{\partial \Sigma}{\partial K}(S_0,K,T) \right] + \underbrace{S_0 \frac{\partial C_{BS}}{\partial S_0}(S_0, K, T; \Sigma) + K \frac{\partial C_{BS}}{\partial K}(S_0,K,T;\Sigma) - C_{BS}(S_0, K, T; \Sigma)}_{=0 \text{ (BS space homogenenity) }} = 0 \end{gather*}

such that the following relationship holds for all space homogeneous models \begin{equation} \frac{\partial \Sigma}{\partial S_0}(S_0,K,T) = -\frac{K}{S_0} \frac{\partial \Sigma}{\partial K}(S_0,K,T) \end{equation}
which is consistent with a sticky moneyness (= sticky delta) behaviour, see below.

[IV stickiness (2/2)] A sticky moneyness (= sticky delta) implied volatility surface is such that $$ \Sigma(S_0+\delta S_0, K, T) = \Sigma(S_0, K^*, T) $$ provided, as the name indicates, that we are working iso-moneyness, meaning that $$\frac{K^*}{S_0} = \frac{K}{S_0+\delta S_0} \iff K^* = K(1 + \delta S_0/S_0)^{-1}$$

Under such circumstances, \begin{align} \frac{\partial \Sigma}{\partial S_0}(S_0, K, T) &= \lim_{\delta S_0 \to 0} \frac{\Sigma(S_0+\delta S_0, K, T) - \Sigma(S_0, K, T)}{\delta S_0} \nonumber \\ &= \lim_{\delta S_0 \to 0} \frac{\Sigma\left(S_0, K(1 + \delta S_0/S_0)^{-1}, T\right) - \Sigma(S_0, K, T)}{\delta S_0} \nonumber \\ &= \lim_{\delta S_0 \to 0} \frac{\Sigma\left(S_0, K(1 - \delta S_0/S_0), T\right) - \Sigma(S_0, K, T)}{\delta S_0} \nonumber \\ &= \lim_{\delta K \to 0} \frac{\Sigma\left(S_0, K-\delta K, T\right) - \Sigma(S_0, K, T)}{\frac{S_0}{K}\delta K} \nonumber\\ &= -\frac{K}{S_0} \frac{\partial \Sigma}{\partial K}(S_0, K, T) \end{align} which is the relationship stemming from price homogeneity we found above.

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  • $\begingroup$ Thank you so much for such a detailed answer. $\endgroup$ – Ussu Oct 28 '19 at 20:38

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